Try Kodak T400CN, quite a popular film for B&W shooters. Also try Ilford Delta, HP5, FP4 and the ever popular Kodak Tri-X. I am quite out of touch with these films already though, but do give them a try.
What B&W film you want to shoot depends a lot on who's going to process and print.
For easy lab processing I'd recommend Kodak T400CN or Ilford XP2. These are C41 films like your colour neg film and can be processed by any minilab.
If you want true B&W, then I'd recommend starting with Ilford HP5+ - it is very easy to process and very difficult to screw up, has a huge tonal range and a lovely grayscale. [I could never get good results from Tri-X ]. You could also try Delta 400 - I get true 400 speed from this film (with ID-11 1:1). If you want a slower film, there's the one and only Ilford PanF+.
I've seen references to HP4 (non-Plus) but that was many years ago. Ilford only makes the plus range of films now (in addition to Delta and the other films) : PanF+, FP4+ and HP5+. I've never used the non-plus range.